Function Operations and Inverses
Lessons in this chapter
About Function Operations and Inverses
Function Operations and Inverses covers building new functions from old ones, and undoing a function altogether.
Performing Function Operations covers adding, subtracting, multiplying, dividing, and composing two functions together to build a new one.
Inverse of a Function closes the chapter by finding a function that exactly reverses another, undoing its input-output pairing.
Both lessons, Performing Function Operations and Inverse of a Function, open as Algebra 2 worksheets worked in the page and scored as you answer. It is a short chapter, so the practice problems read as one sitting rather than a week of work. Free to try, with no login needed for your first 3.
Frequently asked questions
Performing Function Operations covers adding, subtracting, multiplying, and dividing two functions, and composing one with another, including what each of those does to the domain of the result. Inverse of a Function then runs a function backwards, finding the inverse algebraically, checking it by composition, and seeing it as a reflection in the line y = x. This sits in the Functions (F) conceptual category of the high school Common Core.
You can add, subtract, multiply, or divide two functions by combining their output values, or compose them by feeding one function’s output into the other as input.
Function composition, written (f ∘ g)(x) or f(g(x)), means evaluate g first, then use that result as the input to f: order matters, and it usually isn't the same as (g ∘ f)(x).
An inverse function reverses a function’s input-output pairing exactly: if f(a) = b, then the inverse of f maps b back to a.
Swap x and y in the function's equation and solve for the new y: that gives the inverse function's equation.
Not every function has an inverse that's also a function: only a one-to-one function (one that passes the horizontal line test) has a true inverse function.